Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory (including the Diagonal Argument, the Continuum Hypothesis and Cantor’s Theorem) and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing (completed) infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems.

Fuzzy Absolutes and Related Topics

The study on the fuzzy absolutes and related topics. The different kinds of extensions especially compactification formed a major area of study in topology. Perfect continuous mappings always preserve certain topological properties. The concept of Fuzzy sets introduced by the American Cyberneticist L. A Zadeh started a revolution in every branch of knowledge and in particular in every branch of mathematics. Fuzziness is a kind of uncertainty and uncertainty of a symbol lies in the lack of well-defined boundaries of the set of objects to which this symbol belongs. Introduce an s-continuous mapping from a topological space to a fuzzy topological space and prove that the image of an H-closed space under an s-continuous mapping is f-H closed. Here also proved that the arbitrary product fi and sum of  fi of the s-continuous maps fi are also s-continuous. The original motivation behind the study of absolutes was the problem of characterizing the projective objects in the category of compact spaces and continuous functions.

Investigations on the Dynamics of Combination Maps and the Analysis of Bifurcation in a Discontinuous Logistic Map

This thesis is a study of discrete nonlinear systems represented by one dimensional mappings.As one dimensional interative maps represent Poincarre sections of higher dimensional flows,they offer a convenient means to understand the dynamical evolution of many physical systems.It highlighting the basic ideas of deterministic chaos.Qualitative and quantitative measures for the detection and characterization of chaos in nonlinear systems are discussed.Some simple mathematical models exhibiting chaos are presented.The bifurcation scenario and the possible routes to chaos are explained.It present the results of the numerical computational of the Lyapunov exponents (λ) of one dimensional maps.This thesis focuses on the results obtained by our investigations on combinations maps,scaling behaviour of the Lyapunov characteristic exponents of one dimensional maps and the nature of bifurcations in a discontinous logistic map.It gives a review of the major routes to chaos in dissipative systems,namely, Period-doubling ,Intermittency and Crises.This study gives a theoretical understanding of the route to chaos in discontinous systems.A detailed analysis of the dynamics of a discontinous logistic map is carried out, both analytically and numerically ,to understand the route it follows to chaos.The present analysis deals only with the case of the discontinuity parameter applied to the right half of the interval of mapping.A detailed analysis for the n –furcations of various periodicities can be made and a more general theory for the map with discontinuities applied at different positions can be on a similar footing

Study on Fuzzy Ordered FuzzyTopological Spaces

In this study we combine the notions of fuzzy order and fuzzy topology of Chang and define fuzzy ordered fuzzy topological space. Its various properties are analysed. Product, quotient, union and intersection of fuzzy orders are introduced. Besides, fuzzy order preserving maps and various fuzzy completeness are investigated. Finally an attempt is made to study the notion of generalized fuzzy ordered fuzzy topological space by considering fuzzy order defined on a fuzzy subset.

Scaling relations in the Lyapunov exponents of one dimensional maps

We establish numerically the validity of Huberman-Rudnick scaling relation for Lyapunov exponents during the period doubling route to chaos in one dimensional maps. We extend our studies to the context of a combination map. where the scaling index is found to be different.

Financial crisis: theory and practice

In the last 50 years, we have had approximately 40 events with characteristics related to financial crisis. The most severe crisis was in 1929, when the financial markets plummet and the US gross domestic product decline in more than 30 percent. Recently some years ago, a new crisis developed in the United States, but instantly caused consequences and effects in the rest of the world. This new economic and financial crisis has increased the interest and motivation for the academic community, professors and researchers, to understand the causes and effects of the crisis, to learn from it. This is the one of the main reasons for the compilation of this book, which begins with a meeting of a group of IAFI researchers from the University of Barcelona, where researchers form Mexico and Spain, explain causes and consequences of the crisis of 2007. For that reason, we believed this set of chapters related to methodologies, applications and theories, would conveniently explained the characteristics and events of the past and future financial crisis This book consists in 3 main sections, the first one called "State of the Art and current situation", the second named "Econometric applications to estimate crisis time periods" , and the third one "Solutions to diminish the effects of the crisis". The first section explains the current point of view of many research papers related to financial crisis, it has 2 chapters. In the first one, it describe and analyzes the models that historically have been used to explain financial crisis, furthermore, it proposes to used alternative methodologies such as Fuzzy Cognitive Maps. On the other hand , Chapter 2 , explains the characteristics and details of the 2007 crisis from the US perspective and its comparison to 1929 crisis, presenting some effects in Mexico and Latin America. The second section presents two econometric applications to estimate possible crisis periods. For this matter, Chapter 3, studies 3 Latin-American countries: Argentina, Brazil and Peru in the 1994 crisis and estimates the multifractal characteristics to identify financial and economic distress. Chapter 4 explains the crisis situations in Argentina (2001), Mexico (1994) and the recent one in the United States (2007) and its effects in other countries through a financial series methodology related to the stock market. The last section shows an alternative to prevent the effects of the crisis. The first chapter explains the financial stability effects through the financial system regulation and some globalization standards. Chapter 6, study the benefits of the Investor activism and a way to protect personal and national wealth to face the financial crisis risks.

Some problems in set topology relating group of homeomorphisms and order

In this thesis we investigate some problems in set theoretical topology related to the concepts of the group of homeomorphisms and order. Many problems considered are directly or indirectly related to the concept of the group of homeomorphisms of a topological space onto itself. Order theoretic methods are used extensively. Chapter-l deals with the group of homeomorphisms. This concept has been investigated by several authors for many years from different angles. It was observed that nonhomeomorphic topological spaces can have isomorphic groups of homeomorphisms. Many problems relating the topological properties of a space and the algebraic properties of its group of homeomorphisms were investigated. The group of isomorphisms of several algebraic, geometric, order theoretic and topological structures had also been investigated. A related concept of the semigroup of continuous functions of a topological space also received attention

Some Lattice Problems In Fuzzy Set Theory And Fuzzy Topology

It is believed that every fuzzy generalization should be formulated in such a way that it contain the ordinary set theoretic notion as a special case. Therefore the definition of fuzzy topology in the line of C.L.CHANG E9] with an arbitrary complete and distributive lattice as the membership set is taken. Almost all the results proved and presented in this thesis can, in a sense, be called generalizations of corresponding results in ordinary set theory and set topology. However the tools and the methods have to be in many of the cases, new. Here an attempt is made to solve the problem of complementation in the lattice of fuzzy topologies on a set. It is proved that in general, the lattice of fuzzy topologies is not complemented. Complements of some fuzzy topologies are found out. It is observed that (L,X) is not uniquely complemented. However, a complete analysis of the problem of complementation in the lattice of fuzzy topologies is yet to be found out

Analytical Studies on Diffusion and lntermittency in Chaotic Maps

The study of simple chaotic maps for non-equilibrium processes in statistical physics has been one of the central themes in the theory of chaotic dynamical systems. Recently, many works have been carried out on deterministic diffusion in spatially extended one-dimensional maps This can be related to real physical systems such as Josephson junctions in the presence of microwave radiation and parametrically driven oscillators. Transport due to chaos is an important problem in Hamiltonian dynamics also. A recent approach is to evaluate the exact diffusion coefficient in terms of the periodic orbits of the system in the form of cycle expansions. But the fact is that the chaotic motion in such spatially extended maps has two complementary aspects- - diffusion and interrnittency. These are related to the time evolution of the probability density function which is approximately Gaussian by central limit theorem. It is noticed that the characteristic function method introduced by Fujisaka and his co-workers is a very powerful tool for analysing both these aspects of chaotic motion. The theory based on characteristic function actually provides a thermodynamic formalism for chaotic systems It can be applied to other types of chaos-induced diffusion also, such as the one arising in statistics of trajectory separation. It was noted that there is a close connection between cycle expansion technique and characteristic function method. It was found that this connection can be exploited to enhance the applicability of the cycle expansion technique. In this way, we found that cycle expansion can be used to analyse the probability density function in chaotic maps. In our research studies we have successfully applied the characteristic function method and cycle expansion technique for analysing some chaotic maps. We introduced in this connection, two classes of chaotic maps with variable shape by generalizing two types of maps well known in literature.

Design, implementation and application of a generic framework for integrated regional land-use modeling

Land use is a crucial link between human activities and the natural environment and one of the main driving forces of global environmental change. Large parts of the terrestrial land surface are used for agriculture, forestry, settlements and infrastructure. Given the importance of land use, it is essential to understand the multitude of influential factors and resulting land use patterns. An essential methodology to study and quantify such interactions is provided by the adoption of land-use models. By the application of land-use models, it is possible to analyze the complex structure of linkages and feedbacks and to also determine the relevance of driving forces. Modeling land use and land use changes has a long-term tradition. In particular on the regional scale, a variety of models for different regions and research questions has been created. Modeling capabilities grow with steady advances in computer technology, which on the one hand are driven by increasing computing power on the other hand by new methods in software development, e.g. object- and component-oriented architectures. In this thesis, SITE (Simulation of Terrestrial Environments), a novel framework for integrated regional sland-use modeling, will be introduced and discussed. Particular features of SITE are the notably extended capability to integrate models and the strict separation of application and implementation. These features enable efficient development, test and usage of integrated land-use models. On its system side, SITE provides generic data structures (grid, grid cells, attributes etc.) and takes over the responsibility for their administration. By means of a scripting language (Python) that has been extended by language features specific for land-use modeling, these data structures can be utilized and manipulated by modeling applications. The scripting language interpreter is embedded in SITE. The integration of sub models can be achieved via the scripting language or by usage of a generic interface provided by SITE. Furthermore, functionalities important for land-use modeling like model calibration, model tests and analysis support of simulation results have been integrated into the generic framework. During the implementation of SITE, specific emphasis was laid on expandability, maintainability and usability. Along with the modeling framework a land use model for the analysis of the stability of tropical rainforest margins was developed in the context of the collaborative research project STORMA (SFB 552). In a research area in Central Sulawesi, Indonesia, socio-environmental impacts of land-use changes were examined. SITE was used to simulate land-use dynamics in the historical period of 1981 to 2002. Analogous to that, a scenario that did not consider migration in the population dynamics, was analyzed. For the calculation of crop yields and trace gas emissions, the DAYCENT agro-ecosystem model was integrated. In this case study, it could be shown that land-use changes in the Indonesian research area could mainly be characterized by the expansion of agricultural areas at the expense of natural forest. For this reason, the situation had to be interpreted as unsustainable even though increased agricultural use implied economic improvements and higher farmers' incomes. Due to the importance of model calibration, it was explicitly addressed in the SITE architecture through the introduction of a specific component. The calibration functionality can be used by all SITE applications and enables largely automated model calibration. Calibration in SITE is understood as a process that finds an optimal or at least adequate solution for a set of arbitrarily selectable model parameters with respect to an objective function. In SITE, an objective function typically is a map comparison algorithm capable of comparing a simulation result to a reference map. Several map optimization and map comparison methodologies are available and can be combined. The STORMA land-use model was calibrated using a genetic algorithm for optimization and the figure of merit map comparison measure as objective function. The time period for the calibration ranged from 1981 to 2002. For this period, respective reference land-use maps were compiled. It could be shown, that an efficient automated model calibration with SITE is possible. Nevertheless, the selection of the calibration parameters required detailed knowledge about the underlying land-use model and cannot be automated. In another case study decreases in crop yields and resulting losses in income from coffee cultivation were analyzed and quantified under the assumption of four different deforestation scenarios. For this task, an empirical model, describing the dependence of bee pollination and resulting coffee fruit set from the distance to the closest natural forest, was integrated. Land-use simulations showed, that depending on the magnitude and location of ongoing forest conversion, pollination services are expected to decline continuously. This results in a reduction of coffee yields of up to 18% and a loss of net revenues per hectare of up to 14%. However, the study also showed that ecological and economic values can be preserved if patches of natural vegetation are conservated in the agricultural landscape. -----------------------------------------------------------------------

Test for basis-set errors in relativistic Dirac-Fock-Slater calculations with a numerical AO-DFS-basis

A fully relativistic four-component Dirac-Fock-Slater program for diatomics, with numerically given AO's as basis functions is presented. We discuss the problem of the errors due to the finite basis-set, and due to the influence of the negative energy solutions of the Dirac Hamiltonian. The negative continuum contributions are found to be very small.